Two men from South Yorkshire – one with HIV – have admitted dozens of sex crimes against teenage boys across the country.

Anthony Marsh and Lee Davis, both from Doncaster, used social media sites, including the gay sex app Grindr and the now defunct Bebo, to groom 10 boys aged as young as 13 over a period of seven years.

Some of their victims had only just come out as gay and had their first sexual experience with the men, according to the detective in charge of the case. The sex was usually unprotected and Marsh, who has HIV, did not tell the teenagers that he carried the virus.

Police have identified 20 victims so far – all of whom have since tested negative for HIV, the Guardian understands. But detectives fear there could be dozens more young men who had sex with Marsh.

Most of the abused boys willingly met the men and consented to sexual activity, according to Detective Inspector Delphine Waring from South Yorkshire police, who led the inquiry. Marsh, 53, and Davis, 39, did not lie about their ages but knew most of their victims were underage, she said.

"Many of these children were confused about their sexuality or were vulnerable in other ways – some had only just come out at school and were being bullied when they were preyed upon by these men," said Waring. "Initially quite a few of the boys did not see themselves as victims because they had willingly gone to meet Marsh or Davis and there was no coercion or force used in most cases … For some of the boys this was their first sexual experience."

Marsh, a travelling salesman from Hatfield who also used the names Tony Taylor and Tony Smith, has pleaded guilty to 34 offences including possessing and distributing indecent images of children, and sexual activity with a child. He met his victims in hotels when he was travelling around the country for work.

Lee Davies. Photograph: South Yorkshire police

Unemployed Davis, from Conisbrough, who also used the name James Parkin, has admitted 21 offences including raping a 13-year-old boy, sexual activity with a child and taking indecent photographs of a child. A number of his victims were abused at his house.

The men also engaged in sexual activity with the teenagers in their cars and in public spaces and sometimes secretly filmed their encounters.

The two men were not friends offline but admitted sharing the contact details of boys with each other. Both have also pleaded guilty to one joint charge of conspiring to commit sexual activity with children. The offences to which they pleaded guilty date back to as far as 2005 and involved 10 teenage boys aged between 13 and 17 from South Yorkshire, Humberside, the north-east, north-west and the Midlands. Police have since identified 10 more victims, including a 12-year-old boy who was groomed by one of the defendants but never met them offline.

All have received help and support from police and health professionals, said Waring, who warned: "There could be dozens more victims out there because the offending has been happening for such a long time."

The men admitted 55 charges but denied a further 14 offences at a court hearing on 27 September. In Davis's case, he claims he thought some of the boys were over 16.

After the pleas were entered, the Crown Prosecution Service was granted two weeks by the court to decide whether to pursue these remaining charges for trial. On Monday the CPS is to inform the court and South Yorkshire police that it will not seek a trial. The men will now be sentenced on 2 December for the crimes they have admitted.

The investigation began in May 2012 when North Yorkshire police received a tipoff that a man called Tony was viewing indecent images of children on a computer in a hotel room in Scarborough. Marsh's laptop was examined after his arrest and detectives realised that, along with Davis, he had been grooming a large number of teenage boys for sexual activity.

Most of the victims never confided in anyone about meeting Marsh or Davis before the police investigation began, said Waring. She added: "I hope this case raises awareness and serves as a warning to anyone who believes they can abuse young children for their own sexual gratification. I also hope it will encourage parents to monitor their children's internet use.

"In this day and age it's very difficult to stop children using computers, but I would encourage parents and other professionals to be curious and indeed quite intrusive if need be to find out what their children are doing online in order to protect them from online predators."

• This article was amended on 14 October 2013 to correct an error in the headline relating to the number and ages of the victims.